Patent GB-1290339-A (GB)
IPC: G05B23/02
Filed: 1969-06-09
Abstract: 1290339 Root locus plotting NATIONAL RESEARCH DEVELOPMENT CORP 9 Sept 1970 [9 June 1969] 28985/69 Heading G4A An apparatus for plotting root loci showing the variation of the poles of the transfer function of a closed loop system when the gain of the system varies is The system may represent electric voltage or current, hydraulic pressure or the movement of members coupled by mechanical linkages and may be used with a cathode ray oscilloscope in laboratories or teaching for giving rapid root locus displays. The apparatus can be a special purpose digital electric apparatus (Fig. 1) or a computer.  Theory The closed loop transfer function is defined as: where K is gain and F(s) varies as frequency, and can be plotted on an s plane having real and unreal parts using polar co-ordinates R, #. In general: where m # n, and r 1  to r n , q 1  to q m  are constants Where R = # (i.e. KF(s) = #)it is called a pole, when R=0 it is called a zero. Also arg. F(s) = (2k + 1). From these equations it can be proved that poles of the closed loop function occur when where #r is the angle subtended at the point s from the rth zero and #q is the angle subtended at the point 8 from the #q pole. Points in the loci of poles generated when K varies can be obtained from the roots (r 1  to r n  and q 1  to q n ) of F(s) and these are called root loci. Any point on the root locus satisfies this equation: which can be written where # zi  is the angle which a line joining the point to the zero zi makes with the α pj  (real) axis and # pj  is the angle which a line joining the pole Pj makes with the α axis.  The method.-(Fig. 3) The direction of the tangent 68 to the required locus at a pole 63 of the transfer function when open loop is calculated, the phase error (#-#) at a trial point 80 on the tangent is calculated (where phase error is the angle between the tangent and the line joining the pole and the corresponding locus point 82), then, if the phase error is more than the predetermined error, the phase error of a second point lying at a predetermined distance from the pole on a line (whose angle to the tangent equals or is proportional to the phase error at the first point and which is on the same side of the tangent as the required locus) is found, this is repeated for further phase errors (the line directions varying as the sum of the previous phase errors) until the final phase error is less than the predetermined error when the point is plotted. The apparatus.-On Fig. 3 poles 63, 64, 65 of an open loop transfer function are plotted on the s plane, the loci of the corresponding closed loop function being shown at 66, 67. A binary 1 is set in shift registers 41, 42 by logic 43 and potentiometers 11-16 and 21-26 representing the cartesian co-ordinates of 6 (or fewer) poles of the open loop function are connected by analogue switches 32, 33 to circuits 34, 35 and thence to resolver 44. Potentiometers 17-20, 27-30 represent the 4 (or fewer) zeros of the function. Amplifier 36 represents the difference x<SP>1</SP> between the x co-ordinates of first and second poles and amplifier 39 represents the difference y<SP>1</SP> between the y co-ordinates of these poles. These differences are the initial conditions for the simple harmonic motion equation having the solution s = Asinwt + Bcoswt. The resolver 44 includes two integrators and clock pulses pass to counters 51, 53 until comparator 55 detects a zero when counters 51, 53 indicate the angle of the line joining the poles and output 47 (Fig. 2, not shown) indicates the length R of the line. Next the angle of the line joining the first and third poles is computed and counters 51, 53 pulsed to provide a cumulative total of the angle and this is repeated for seven poles when the procedure is reversed and subtraction occurs for seven angles for the angles of lines joining the poles to the zeros, thus equation (4) is computed by subtracting # and counters 51, 53 contain the phase error angle #. Using counter 52 the distance As is applied and the coordinates of the first trial point 80 produced with reference to the first pole 63, and subsequently turned into true co-ordinates. Then a similar process to that for 0 is carried out for each trial point error (including a phase error # 1 ) until the phase error in counter 53 is below a predetermined value when the point on the locus is read into circuits 60, 61 and thence to the oscilloscope. A new trial point is produced from the data of the last point and the process repeated. The gain of the system at the trial point is calculated by multiplying together the lengths of all lines joining the trial point and the poles by adding logarithms using logarithmic amplifier 56 and dividing this by the product of the lengths of lines joining the trial point and the zeros. The whole process is constantly repeated to give a fresh CRO trace. If the rate of change of phase error is large the number of calculations involved may be excessive and addition of a counter 70 can be used to reduce this. If the value of k at any locus point is less than that at the previous point a break point has occurred (i.e. the locus has changed direction by 90‹). These are detected by circuits 71, 72. If the open loop transfer has a pure time delay of T seconds (i.e. caused by an operator's delay) then: A time delay -Y(T) can be accounted for by passing a count to counter 52 from amplifier 86 by way of analog to digital converter 87.